The Black-Litterman model begins with a neutral starting point of ``equilibrium risk premiums". It can be calculated by:
\begin{equation}
\Pi = \lambda \Sigma w_{mkt}
\end{equation}
where $\Pi$ is the implied excess equilibrium return vector ($N \times 1$), $\lambda$ is the risk aversion coefficient, $\Sigma$ is the covariance matrix ($N \times N$) of excess returns, and $w_{mkt}$ is the market capitalization weights of each portfolio ($N \times 1$). 

The covariance matrix $\Sigma$ is evaluated using the training data (in the expanding or rolling window). To compute the $w_{mkt}$, we use the average firm size and total number of firms for the 6 portfolios:
% comment: I'm not sure how much I like the clarity of the 1-6 Summation. Something summing over set 
% notation is probably preferred.

\begin{equation}
w_{i, mkt} = \frac{msize_i \times nfirms_i}{\sum_1^6{msize_i \times nfirms_i}}
\end{equation}

The risk-aversion coefficient ($\lambda$) describes the tradeoff of expected risk-return. It models how much the collective market of investors will forego expected return in exchange for less variance. The market implied risk-aversion coefficient ($\lambda$) for a portfolio can be estimated by dividing the expected excess return by the portfolio variance:
\begin{equation}
\lambda = \frac{E(r) - r_f}{\sigma^2}
\end{equation}

% comment: this is indenting but its not a new paragraph. Can that be disabled?

where $E(r)$ is the total return of the expected market or benchmark, $r_f$ is the risk-free rate, and $\sigma^2 = w_{mkt}^T\Sigma w_{mkt}$ is the variance of the market or benchmark excess returns. To compute the risk-aversion coefficient, we use the S\&P500 index monthly returns (pulled from Yahoo! Finance) as our benchmark, and set risk-free rate to 0. With these assumptions, we get $\lambda = 2.52$ as our risk aversion coefficient. It is important to note that we matched end-of-month daily S\&P500 returns to create monthly returns instead of relying on the more easily available monthly daily captured on the first of the month. This is a critical step to avoid look ahead bias. Our choice of a hypothetically 0 risk-free rate is merely a simplification of our model. All of our code base supports the introduction of a non-zero risk-free rate.

% write more here about the choice of matching equity investment horizons to interest products

Table \ref{table:prima} shows the comparison between market equilibrium prima vs. historical prima at one sample period. As we noticed, that the market equilibrium prima has a smaller estimates than historical prima. 

\begin{table*}[htbp]
   \centering
   \begin{tabular}{@{} cccccccc @{}} % Column formatting, @{} suppresses leading/trailing space
   \hline
   	&Date			& SMLO		&  SMME 		& SMHI  & BILO &BIME & BIHI\\
    \hline
Market Equilibrium Prima & 2012-01-31 &0.00612 & 0.00498 & 0.00521 &0.00465 &0.00476 &0.00495\\
Historical Prima  & 2012-01-31& 0.07631 &  0.06597 & 0.05666 & 0.05827 & 0.03633 & 0.03614\\
   \hline
   \end{tabular}
   \caption{Comparison of Market Equilibrium Prima vs. Historical Prima (log returns) at one sample period}
   \label{table:prima}
\end{table*}
